The New International Bridge (Nuevo Puente International) is the easiest way to get from the United States into Matamoros. Individuals may drive their automobiles across, but most U.S. insurance companies do not cover you in Mexico, and Mexican insurance is required by law. Only an idiot would try to avoid it. Insurance companies are located on both sides of the border to help you with the details.
For us, and for thousands of other short-term visitors, the more reasonable option is to park your car and take the short walk over the bridge. A couple of large parking lots are located just a block or so west of the bridge, and all day parking was only $4.00.
Immediately after crossing the New International Bridge into Matamoros you will see a large bus stop. From here one can catch a bus to anywhere in the city, or the entire country. A fundamental knowledge of Spanish would be helpful, as most of the bus drivers are not bilingual. However, by asking around you can usually find somone eager to help you. We stepped into a shop just over the border and acquired a city street map, then hopped onto a bus for El Mercado (The Market), about two miles away. From the market we walked over much of the city, to the Cathedral, the Casamata Museum, and back to the border. That meant three or four miles of walking through the hot streets of the city, but it also allowed us to see much more than we would have otherwise.